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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.solvers;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.exception.NoBracketingException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.TooManyEvaluationsException;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    /**<a name="line.24"></a>
<FONT color="green">025</FONT>     * This class implements the &lt;a href="http://mathworld.wolfram.com/MullersMethod.html"&gt;<a name="line.25"></a>
<FONT color="green">026</FONT>     * Muller's Method&lt;/a&gt; for root finding of real univariate functions. For<a name="line.26"></a>
<FONT color="green">027</FONT>     * reference, see &lt;b&gt;Elementary Numerical Analysis&lt;/b&gt;, ISBN 0070124477,<a name="line.27"></a>
<FONT color="green">028</FONT>     * chapter 3.<a name="line.28"></a>
<FONT color="green">029</FONT>     * &lt;p&gt;<a name="line.29"></a>
<FONT color="green">030</FONT>     * Muller's method applies to both real and complex functions, but here we<a name="line.30"></a>
<FONT color="green">031</FONT>     * restrict ourselves to real functions.<a name="line.31"></a>
<FONT color="green">032</FONT>     * This class differs from {@link MullerSolver} in the way it avoids complex<a name="line.32"></a>
<FONT color="green">033</FONT>     * operations.&lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * Except for the initial [min, max], it does not require bracketing<a name="line.34"></a>
<FONT color="green">035</FONT>     * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex<a name="line.35"></a>
<FONT color="green">036</FONT>     * number arises in the computation, we simply use its modulus as real<a name="line.36"></a>
<FONT color="green">037</FONT>     * approximation.&lt;/p&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     * &lt;p&gt;<a name="line.38"></a>
<FONT color="green">039</FONT>     * Because the interval may not be bracketing, bisection alternative is<a name="line.39"></a>
<FONT color="green">040</FONT>     * not applicable here. However in practice our treatment usually works<a name="line.40"></a>
<FONT color="green">041</FONT>     * well, especially near real zeroes where the imaginary part of complex<a name="line.41"></a>
<FONT color="green">042</FONT>     * approximation is often negligible.&lt;/p&gt;<a name="line.42"></a>
<FONT color="green">043</FONT>     * &lt;p&gt;<a name="line.43"></a>
<FONT color="green">044</FONT>     * The formulas here do not use divided differences directly.&lt;/p&gt;<a name="line.44"></a>
<FONT color="green">045</FONT>     *<a name="line.45"></a>
<FONT color="green">046</FONT>     * @version $Id: MullerSolver2.java 1379560 2012-08-31 19:40:30Z erans $<a name="line.46"></a>
<FONT color="green">047</FONT>     * @since 1.2<a name="line.47"></a>
<FONT color="green">048</FONT>     * @see MullerSolver<a name="line.48"></a>
<FONT color="green">049</FONT>     */<a name="line.49"></a>
<FONT color="green">050</FONT>    public class MullerSolver2 extends AbstractUnivariateSolver {<a name="line.50"></a>
<FONT color="green">051</FONT>    <a name="line.51"></a>
<FONT color="green">052</FONT>        /** Default absolute accuracy. */<a name="line.52"></a>
<FONT color="green">053</FONT>        private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;<a name="line.53"></a>
<FONT color="green">054</FONT>    <a name="line.54"></a>
<FONT color="green">055</FONT>        /**<a name="line.55"></a>
<FONT color="green">056</FONT>         * Construct a solver with default accuracy (1e-6).<a name="line.56"></a>
<FONT color="green">057</FONT>         */<a name="line.57"></a>
<FONT color="green">058</FONT>        public MullerSolver2() {<a name="line.58"></a>
<FONT color="green">059</FONT>            this(DEFAULT_ABSOLUTE_ACCURACY);<a name="line.59"></a>
<FONT color="green">060</FONT>        }<a name="line.60"></a>
<FONT color="green">061</FONT>        /**<a name="line.61"></a>
<FONT color="green">062</FONT>         * Construct a solver.<a name="line.62"></a>
<FONT color="green">063</FONT>         *<a name="line.63"></a>
<FONT color="green">064</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.64"></a>
<FONT color="green">065</FONT>         */<a name="line.65"></a>
<FONT color="green">066</FONT>        public MullerSolver2(double absoluteAccuracy) {<a name="line.66"></a>
<FONT color="green">067</FONT>            super(absoluteAccuracy);<a name="line.67"></a>
<FONT color="green">068</FONT>        }<a name="line.68"></a>
<FONT color="green">069</FONT>        /**<a name="line.69"></a>
<FONT color="green">070</FONT>         * Construct a solver.<a name="line.70"></a>
<FONT color="green">071</FONT>         *<a name="line.71"></a>
<FONT color="green">072</FONT>         * @param relativeAccuracy Relative accuracy.<a name="line.72"></a>
<FONT color="green">073</FONT>         * @param absoluteAccuracy Absolute accuracy.<a name="line.73"></a>
<FONT color="green">074</FONT>         */<a name="line.74"></a>
<FONT color="green">075</FONT>        public MullerSolver2(double relativeAccuracy,<a name="line.75"></a>
<FONT color="green">076</FONT>                            double absoluteAccuracy) {<a name="line.76"></a>
<FONT color="green">077</FONT>            super(relativeAccuracy, absoluteAccuracy);<a name="line.77"></a>
<FONT color="green">078</FONT>        }<a name="line.78"></a>
<FONT color="green">079</FONT>    <a name="line.79"></a>
<FONT color="green">080</FONT>        /**<a name="line.80"></a>
<FONT color="green">081</FONT>         * {@inheritDoc}<a name="line.81"></a>
<FONT color="green">082</FONT>         */<a name="line.82"></a>
<FONT color="green">083</FONT>        @Override<a name="line.83"></a>
<FONT color="green">084</FONT>        protected double doSolve()<a name="line.84"></a>
<FONT color="green">085</FONT>            throws TooManyEvaluationsException,<a name="line.85"></a>
<FONT color="green">086</FONT>                   NumberIsTooLargeException,<a name="line.86"></a>
<FONT color="green">087</FONT>                   NoBracketingException {<a name="line.87"></a>
<FONT color="green">088</FONT>            final double min = getMin();<a name="line.88"></a>
<FONT color="green">089</FONT>            final double max = getMax();<a name="line.89"></a>
<FONT color="green">090</FONT>    <a name="line.90"></a>
<FONT color="green">091</FONT>            verifyInterval(min, max);<a name="line.91"></a>
<FONT color="green">092</FONT>    <a name="line.92"></a>
<FONT color="green">093</FONT>            final double relativeAccuracy = getRelativeAccuracy();<a name="line.93"></a>
<FONT color="green">094</FONT>            final double absoluteAccuracy = getAbsoluteAccuracy();<a name="line.94"></a>
<FONT color="green">095</FONT>            final double functionValueAccuracy = getFunctionValueAccuracy();<a name="line.95"></a>
<FONT color="green">096</FONT>    <a name="line.96"></a>
<FONT color="green">097</FONT>            // x2 is the last root approximation<a name="line.97"></a>
<FONT color="green">098</FONT>            // x is the new approximation and new x2 for next round<a name="line.98"></a>
<FONT color="green">099</FONT>            // x0 &lt; x1 &lt; x2 does not hold here<a name="line.99"></a>
<FONT color="green">100</FONT>    <a name="line.100"></a>
<FONT color="green">101</FONT>            double x0 = min;<a name="line.101"></a>
<FONT color="green">102</FONT>            double y0 = computeObjectiveValue(x0);<a name="line.102"></a>
<FONT color="green">103</FONT>            if (FastMath.abs(y0) &lt; functionValueAccuracy) {<a name="line.103"></a>
<FONT color="green">104</FONT>                return x0;<a name="line.104"></a>
<FONT color="green">105</FONT>            }<a name="line.105"></a>
<FONT color="green">106</FONT>            double x1 = max;<a name="line.106"></a>
<FONT color="green">107</FONT>            double y1 = computeObjectiveValue(x1);<a name="line.107"></a>
<FONT color="green">108</FONT>            if (FastMath.abs(y1) &lt; functionValueAccuracy) {<a name="line.108"></a>
<FONT color="green">109</FONT>                return x1;<a name="line.109"></a>
<FONT color="green">110</FONT>            }<a name="line.110"></a>
<FONT color="green">111</FONT>    <a name="line.111"></a>
<FONT color="green">112</FONT>            if(y0 * y1 &gt; 0) {<a name="line.112"></a>
<FONT color="green">113</FONT>                throw new NoBracketingException(x0, x1, y0, y1);<a name="line.113"></a>
<FONT color="green">114</FONT>            }<a name="line.114"></a>
<FONT color="green">115</FONT>    <a name="line.115"></a>
<FONT color="green">116</FONT>            double x2 = 0.5 * (x0 + x1);<a name="line.116"></a>
<FONT color="green">117</FONT>            double y2 = computeObjectiveValue(x2);<a name="line.117"></a>
<FONT color="green">118</FONT>    <a name="line.118"></a>
<FONT color="green">119</FONT>            double oldx = Double.POSITIVE_INFINITY;<a name="line.119"></a>
<FONT color="green">120</FONT>            while (true) {<a name="line.120"></a>
<FONT color="green">121</FONT>                // quadratic interpolation through x0, x1, x2<a name="line.121"></a>
<FONT color="green">122</FONT>                final double q = (x2 - x1) / (x1 - x0);<a name="line.122"></a>
<FONT color="green">123</FONT>                final double a = q * (y2 - (1 + q) * y1 + q * y0);<a name="line.123"></a>
<FONT color="green">124</FONT>                final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;<a name="line.124"></a>
<FONT color="green">125</FONT>                final double c = (1 + q) * y2;<a name="line.125"></a>
<FONT color="green">126</FONT>                final double delta = b * b - 4 * a * c;<a name="line.126"></a>
<FONT color="green">127</FONT>                double x;<a name="line.127"></a>
<FONT color="green">128</FONT>                final double denominator;<a name="line.128"></a>
<FONT color="green">129</FONT>                if (delta &gt;= 0.0) {<a name="line.129"></a>
<FONT color="green">130</FONT>                    // choose a denominator larger in magnitude<a name="line.130"></a>
<FONT color="green">131</FONT>                    double dplus = b + FastMath.sqrt(delta);<a name="line.131"></a>
<FONT color="green">132</FONT>                    double dminus = b - FastMath.sqrt(delta);<a name="line.132"></a>
<FONT color="green">133</FONT>                    denominator = FastMath.abs(dplus) &gt; FastMath.abs(dminus) ? dplus : dminus;<a name="line.133"></a>
<FONT color="green">134</FONT>                } else {<a name="line.134"></a>
<FONT color="green">135</FONT>                    // take the modulus of (B +/- FastMath.sqrt(delta))<a name="line.135"></a>
<FONT color="green">136</FONT>                    denominator = FastMath.sqrt(b * b - delta);<a name="line.136"></a>
<FONT color="green">137</FONT>                }<a name="line.137"></a>
<FONT color="green">138</FONT>                if (denominator != 0) {<a name="line.138"></a>
<FONT color="green">139</FONT>                    x = x2 - 2.0 * c * (x2 - x1) / denominator;<a name="line.139"></a>
<FONT color="green">140</FONT>                    // perturb x if it exactly coincides with x1 or x2<a name="line.140"></a>
<FONT color="green">141</FONT>                    // the equality tests here are intentional<a name="line.141"></a>
<FONT color="green">142</FONT>                    while (x == x1 || x == x2) {<a name="line.142"></a>
<FONT color="green">143</FONT>                        x += absoluteAccuracy;<a name="line.143"></a>
<FONT color="green">144</FONT>                    }<a name="line.144"></a>
<FONT color="green">145</FONT>                } else {<a name="line.145"></a>
<FONT color="green">146</FONT>                    // extremely rare case, get a random number to skip it<a name="line.146"></a>
<FONT color="green">147</FONT>                    x = min + FastMath.random() * (max - min);<a name="line.147"></a>
<FONT color="green">148</FONT>                    oldx = Double.POSITIVE_INFINITY;<a name="line.148"></a>
<FONT color="green">149</FONT>                }<a name="line.149"></a>
<FONT color="green">150</FONT>                final double y = computeObjectiveValue(x);<a name="line.150"></a>
<FONT color="green">151</FONT>    <a name="line.151"></a>
<FONT color="green">152</FONT>                // check for convergence<a name="line.152"></a>
<FONT color="green">153</FONT>                final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy);<a name="line.153"></a>
<FONT color="green">154</FONT>                if (FastMath.abs(x - oldx) &lt;= tolerance ||<a name="line.154"></a>
<FONT color="green">155</FONT>                    FastMath.abs(y) &lt;= functionValueAccuracy) {<a name="line.155"></a>
<FONT color="green">156</FONT>                    return x;<a name="line.156"></a>
<FONT color="green">157</FONT>                }<a name="line.157"></a>
<FONT color="green">158</FONT>    <a name="line.158"></a>
<FONT color="green">159</FONT>                // prepare the next iteration<a name="line.159"></a>
<FONT color="green">160</FONT>                x0 = x1;<a name="line.160"></a>
<FONT color="green">161</FONT>                y0 = y1;<a name="line.161"></a>
<FONT color="green">162</FONT>                x1 = x2;<a name="line.162"></a>
<FONT color="green">163</FONT>                y1 = y2;<a name="line.163"></a>
<FONT color="green">164</FONT>                x2 = x;<a name="line.164"></a>
<FONT color="green">165</FONT>                y2 = y;<a name="line.165"></a>
<FONT color="green">166</FONT>                oldx = x;<a name="line.166"></a>
<FONT color="green">167</FONT>            }<a name="line.167"></a>
<FONT color="green">168</FONT>        }<a name="line.168"></a>
<FONT color="green">169</FONT>    }<a name="line.169"></a>




























































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